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Maths in a minute - combinatorics

And while we're on the topic of probability, let's answer one of those important mathematical question: how likely are you to win the lottery?

In the UK lottery you have to choose 6 numbers out of 49, and for a chance at the jackpot you need all of your 6 numbers to come up in the main draw. So the question is really how many possible combinations of 6 numbers can be drawn out of 49? There are 49 possibilities for the first number, 48 for the second, and so on to 44 possibilities for the sixth number, so there are 49 x 48 x 47 x
46 x 45 x 44 = 10068347520 ways of choosing those six numbers... in that order. But we don't care which order our numbers are picked, and the number of different ways of picking 6 numbers are 6 x 5 x 4 x 3 x 2 x 1 = 6! = 720. Therefore our six numbers are one of 49 x 48 x 47 x 46 x 45 x 44 / 6! = 13983816 so we have about a one in 14 million chance of hitting the jackpot.
Hmmm...

But on a brighter note, we have just discovered a very useful mathematical fact: the number of combinations of size k (sets of objects in which order doesn't matter) from a larger set of size n is n! / (n-k)! / k!.

This sort of argument lies at the heart of combinatorics, the mathematics of counting. It might not help you win lotto, but it might keep you healthy. It is used to understand how viruses such as influenza reproduce and mutate, by assessing the chances of creating viable viruses from random recombination of genetic segments.

You can read more on combinatorics, including money (lotto), love (well kissing frogs) and fun (juggling and rubiks cubes) on Plus.

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Maths Inspiration event in Cambridge — November 24th

There are still places left at the Maths Inspiration morning event in Cambridge on 24th November, organised by Rob Eastaway. All Maths Inspiration events are aimed at sixth formers and more able Year 11 students.

The Cambridge events will be held in West Road Concert Hall on 24th November 2009, 9.45am - 12.30, repeated 1pm - 3.45pm. Speakers include Claire Ellis, Professor Chris Budd, Dr Hugh Hunt and Professor David Spiegelhalter, with talks including What have mathematicians ever done for us?, The Maths of breakfast, and The subtle science of uncertainty.

Tickets are £6 per attendee at all events with one teacher entitled to free entry for every ten students.

For full details of all events and to book tickets, please see the Maths Inspiration website, or contact mail@mathsinspiration.com.

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What would you like to know about your Universe — The last online poll

This poll is now closed. The most popular question was: "How long is a day?" We will publish the answer in an article on Plus shortly. Thank you for taking part!

This is the last online poll in our series to celebrate the International Year of Astronomy 2009. Choose your favourite question from the list on the right, and we'll put the one that proves most popular to world-leading astronomers and cosmologists. The poll will remain open for a month and the answer will be published in a Plus article and
possibly a podcast soon after.

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Cancer is one of the major causes of death in the world (particularly the developed world), with around 11 million people diagnosed and around 7 million people dying each year. The World Health Organisation predicts that current trends show around 9 million will die in 2015, with the number rising to 11.5 million in 2030.

Cancer is the focus of much medical research, but perhaps surprisingly, mathematical research is also playing its part. Mathematician Mark Chaplain and an interdisciplinary team at the University of Dundee, have been awarded 1.7 million euros to develop a virtual model of cancer growth and spread.

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Maths student poster competition just launched!

If you're a maths student and feel like enthusing the next generation with your favourite bit of maths, then why not take part in the Further Mathematics Support Programme (FMSP) and Rolls–Royce plc third national poster competition.

Undergraduate or PGCE mathematics students are invited to design a poster, individually or as a group, which conveys the essence of a mathematical topic that has been covered at university by the designer(s). The poster should be targeted at school and college students studying AS or A level mathematics. Two winners will be chosen and receive £100 each, and their designs will be printed and
sent out to over 2000 schools and colleges. The closing date is the 31st of March 2010. See the FMSP website for previous winners.

If this sounds interesting, then here are the rules. The poster should be mathematically accurate, attractively laid out, capable of enriching a course in AS or A level mathematics, and likely to attract school/college students to take mathematics (or mathematics-related subjects) at university. You can design your poster in any readily available software. Ideally, the page layout should be
set to 59.4cm × 42.0cm, using either landscape or portrait format. The university's logo should appear in the top left corner and there should be a space 7cm high x 5cm wide in the top right corner for the FMSP logo. The bottom 2cm of the poster should be left blank. All images should be at least 300dpi.

Entries, as well as any questions about the competition, should be emailed to Richard Browne at RichardBrowne@furthermaths.org.uk. The email must include the name(s) and full contact details of the designer(s). The poster design should be attached to the e-mail, in the form of an editable file. The FMSP reserves the right to edit the winning designs before printing.

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Michael Green replaces Stephen Hawking in Lucasian chair

Congratulations to Michael Green who has been elected the 18th holder of the Lucasian Professorship of Mathematics at the University of Cambridge, one of the most prestigious academic positions in the mathematical sciences.

Isaac Newton was the second person to hold it in the 17th century, and he has been succeeded by mathematical giants including Charles Babbage, father of computer science, the theoretical physicist Paul Dirac, and of course Stephen Hawking, who has been holding the chair since 1979. Hawking stepped down from the professorship in the year of his 67th birthday, as university statutes require.

Michael Green is one of the founders of string theory, which he pioneered from the early seventies onwards. Apart from original research in the area, his contributions include the a textbook co-authored with Edward Witten and John Schwarz, which for many years remained the only string theory text book around.

Peter Haynes, Head of the Department of Applied Mathematics and Theoretical Physics at Cambridge, said: "Michael Green has played a leading role in theoretical physics research in the department since 1993. He is internationally known as a pioneer in string theory which over the last 20 years has become one of the most important and active areas of the
field. In the department he continues to make important advances in this topic and at the same time to support and inspire young researchers. His appointment as Lucasian Professor continues the very distinguished tradition of that post."